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Let K be a quadratic extension of Q, B a quaternion algebra over Q and A = B ⊗ Q K. Let O be a maximal order in A extending an order in B. The projective norm one group PO 1 is shown to be isomorphic to the spinorial kernel group O'(L), for an explicitly determined quadratic Z-lattice L of rank four, in several general situations. In other cases, only the local structures of O and L are given at each prime. Both definite and indefinite lattices are covered. Some results for quadratic global field extensions K/F and maximal S-orders are given. There is a description of the F-quaternion subalgebras of A, and also of their norm one groups as stabilizer subgroups and as unitary groups. Conjugacy classes of the Fuchsian subgroups of PO 1 corresponding to stabilizer subgroups are studied.

Quaternion algebras, arithmetic Kleinian groups andZ-lattices